All the computer components are connected to each other on a motherboard, also referred to as the main board. All motherboards have special chips that allows each component to communicate with each other components (such as between the processor and the hard drive), as well as to the outside world (USB ports, on-board graphics and sound). The motherboard has connectors that allow you to connect components to the board, such as AGP and PCIE graphics slots for graphics cards, memory slots, the CPU slot, connectors for the hard drives and disk drives, and so on.
Since there are a variety of different processors, memory, graphics cards and drives, each motherboard must be checked to see if it is fully compatible with the components you would like to use.
Discrete motherboards have no "on-board graphics" but do have open graphics slots so that you may add your own graphics card or cards.
Integrated motherboards do have "on-board graphics" and do not require an additional graphics card, and some of these Integrated main boards also have the option to upgrade with a graphics card.